Spatial Pair-Copula Model of Grade for an Anisotropic Gold Deposit

Abstract

Copulas provide a convenient way to express multivariate distribution. In this study, pair-copula modelling is applied to a gold deposit in Western Ghana. The dataset for the gold deposit has 1500 surface soil samples on an area of 7810 ha. The distribution of the grade appears to be anisotropic and has a non-stationary random process. The objectives of the analysis are to use a spatial pair-copula to model the anisotropic gold grade and to determine regions of highest gold value within the field for a future drilling campaign. In the analysis, possible transformations of the data were compared to reduce the influence of outliers and in the case of copulas achieving a marginal uniform distribution. The anisotropy of the gold grades is described with empirical copula contour plots for each distance class and for two orthogonal directions. The non-stationarity is modelled by regression methods. The residuals from the regression are modelled with both spatial pair copulas and kriging. The different approaches are compared in terms of the mean absolute error (MAE) and root mean square error, using different proportions of the data for training and validation. The pair-copula median with kernel margins had MAE of 17.4 compared to 18.3 for log-normal kriging. An advantage of copulas is that they provide a more accurate model than kriging for the uncertainty associated with predictions.

Notes

Acknowledgements

This research is supported by Australian Government Research Training Program Scholarship awarded to Mr. Emmanuel Addo Jr. The authors will like to thank the mining company for providing the surface soil sample datasets used in this case-study. The authors will like express their gratitude to the reviewers for their comments and suggestions, which have improved the practical application of this manuscript.

Supplementary material

Appendix A

Constructing the Vine Copula

The aim of this appendix is to explain the concept of pair-copula models in the context of spatial distributions. The general principles can be demonstrated with a trivariate copula for modelling grade at three locations. There are three stages: defining the model; fitting the model; and making predictions from the model.

Defining the Model

A trivariate pdf for the grades at three locations \( f\left( {z_{1} ,z_{2} ,z_{3} } \right) \) can be factorized as using the definition of conditional probability, and the corresponding multiplicative rule of probability

This is the justification for Eq. (4a) and the corresponding diagram in (Fig. 1a)

In general, the marginal densities \( f\left( {x_{1} } \right), f\left( {x_{2} } \right)\;{\text{and}}\;f\left( {x_{3} } \right) \) are different but in the spatial application there is only one marginal distribution of grade and \( f\left( \right) \) is the common distribution. The copulas \( c_{12} \) and \( c_{13} \) are defined by their type and values of their parameters. The conditional copula \( c_{{\left. {23} \right|1}} \) is also defined by its type and the value(s) of its parameter(s). However, the arguments of the conditional copula are also conditional on \( z_{1} \) , but they can be expressed in terms of unconditional copulas and this is key to the pair-copula construction. The key result is

Joe (1996) proves this result in a general case of multiple conditioning variables. The conditional cumulative distribution functions required for generating the density of the full copula are calculated using the partial derivatives of all the copulas involved. The set of indices of the conditional variables are denoted by \( v \) and the set of indices excluding \( j \) by \( - j \)

Fitting the Copula

The choice of a suitable marginal distribution of grade can be based on the histogram of the grades, or some transformation of grades. If there are many data an empirical kernel smoother may be preferable to some parametric distribution.

The two variable copulas that are considered for the pair-copula construction all have a single parameter that determines the strength of association. The parameter can be expressed as a function of Kendall’s tau and the methods of moments estimate of the parameter is obtained by substituting the sample estimate of Kendall’s tau. In the case of the unconditional copulas, all possible pairs of grades were divided into distance classes and Kendall’s tau was estimated within distance class. These estimates can be interpolated to give an estimate of Kendall’s tau for any distance separating the two points.

The value of Kendall’s tau for the conditional copula can be obtained as follows. Within each distance class consider all possible pairs of pairs with a common location, that is, of the form \( \{ (z_{2} ,z_{1} ), \, (z_{3} ,z_{1} )\} \), and for each pair of pairs construct a pair \( (z_{2} ,z_{3} ) \) which is conditioned on \( z_{1} \). The distances between \( z_{2} \) and \( z_{1} \) and between \( z_{3} \) and \( z_{1} \) are approximately equal as they are in the same distance class, but the distance between \( z_{2} \) and \( z_{3} \) is not so constrained. So the \( (z_{2} ,z_{3} ) \) pairs can be classed by the distance between \( z_{2} \) and \( z_{3} \). Then Kendall’s tau can be calculated within these classes, and hence interpolated for any distances between \( z_{2} \) and \( z_{3} \) for a given category of \( z_{1} \).

Predictions

The fitted model is defined in terms of: the fitted marginal distribution, the forms of the copulas and the parameters for each copula which are obtained from the estimated Kendall’s tau, which in turn depends on the distances between the three points. The aim of fitting the copula is to predict grade at one of the points given numerical values of grade at the other two. The purpose of the prediction may be to: predict grade at an unknown point in the region; predict grade at a point slightly outside the region; or to predict grade at a point where the grade is known so as to calculate a prediction error and compare prediction strategies. The method is the same in all three cases. The fitted copula is the trivariate distribution \( f\left( {\left. {z_{3} } \right|z_{1} ,z_{2} } \right) \) and the conditional distribution of \( z_{3} \) is required, say, on \( z_{1} \) and \( z_{2} \), \( f\left( {z_{1} ,z_{2} ,z_{3} } \right) \) and then the prediction is the mean, or possibly median of this distribution. The Metropolis–Hastings (M–H) algorithm (e.g., Chib and Greenberg 1995) provides a neat solution for finding the mean or median. The M–H algorithm is used to make random draws from \( f\left( {\left. {z_{3} } \right|z_{1} ,z_{2} } \right) \) and so build up the conditional distribution as the distribution of these draws. The mean, median and prediction intervals can all be calculated from this conditional distribution. It is easiest to sample \( F\left( {z_{3} } \right) \) from the pair-copula and then apply \( F^{ - 1} \) to obtain the corresponding conditional distribution of \( z_{3} \). As for any conditional distribution \( c\left( {\left. {F\left( {z_{3} } \right)} \right|F\left( {z_{1} } \right),F\left( {z_{2} } \right)} \right) \) is proportional to \( c\left( {F\left( {z_{1} } \right),F\left( {z_{2} } \right),F\left( {z_{3} } \right)} \right) \) with \( F\left( {z_{2} } \right) \) and \( F\left( {z_{3} } \right) \) replaced by their numerical values. Is referred to as \( c\left( {F\left( {z_{1} } \right),F\left( {z_{2} } \right),F\left( {z_{3} } \right)} \right) \) with numeric values substituted for \( F\left( {z_{2} } \right) {\text{and }}F\left( {z_{3} } \right) \) as \( g\left( w \right) \), where \( w = F\left( {z_{3} } \right) \). A great advantage of the M–H algorithm is that the constant of proportionality (normalizing factor) is not needed. A suitable M–H algorithm implementation is